We find scaling limits for the sizes of the largest components at criticality for rank-1 inhomogeneous random graphs with power-law degrees with power-law exponent ?. We investigate the case where ??(3,4), so that the degrees have finite variance but infinite third moment. The sizes of the largest clusters, rescaled by n?(??2)/(??1), converge to hitting times of a ?thinned? Lévy process, a special case of the general multiplicative coalescents studied by Aldous [Ann. Probab. 25 (1997) 812?854] and Aldous and Limic [Electron. J. Probab. 3 (1998) 1?59].